Cartel Stability, Mark-Up
Cyclicality and Government
Spending Multipliers
Luca Lambertini
Luigi Marattin
Cartel Stability, Mark-Up Cyclicality and
Government Spending Multipliers
Luca Lambertini and Luigi Marattin
Department of Economics, University of Bologna
Strada Maggiore 45, 40125 Bologna, Italy
luca.lambertini@unibo.it; luigi.marattin@unibo.it
March 23, 2012
Abstract
Mark-up cyclical behaviour is relevant in determining the size of government spending multiplier on output. While theoretical liter-ature priviliged the counteryclical hypothesis, empirical evidence is far from being conclusive. Based on seminal Rotemberg and Sa-loner (1986) contribution, we build a theoretical framework based on Bertrand duopoly, stochastic demand and product di¤erentiation, where the analysis of cartel stability under partial collusion points towards procyclical pricing. According to the intensity of marginal cost cyclicality, this can produce a procyclical mark up or - at least - render it less countercyclical than expected, with relevant e¤ects on the transmission mechanism of government spending stimuli.
JEL Codes: C73, L13
Keywords: partial collusion, cyclical pricing
We would like to thank Giacomo Calzolari, Paolo Manasse and Antonio Minniti per precious comments and suggestions. The usual disclaimer applies.
1
Introduction
Recent theoretical contributions (Hall, 2009; Woodford, 2011) highlight the importance of price mark-up’s cyclical behaviour for the transmission
mech-anism of …scal policy. Previous literature (Galì et al., 2005; Galì, 2005)
had already stressed that an exogenous reduction in the aggregate ine¢ -ciency wedge (price or wage mark-up) ampli…es the e¤ects of a government
spending stimulus on aggregate demand, and vice versa. However, by
re-lating mark-up movements to the business cycle, it is possible to investigate analytically the relationship between government spending multipliers and the degree of pro/countercyclicality of mark-up. In a stylized sticky prices
macroeconomic model, Hall shows that if we de…ne (y) = y ! as the price
mark-up, and parameter ! indicates the sensitivity to the income level y, then: sign 2 4@ dy dg @! 3 5 = sign (!)
Considering that the government spending multiplier is positive, equation (1) means that if the mark-up is countercyclical (! > 0) then the higher the sensitivity to aggregate demand (! "), the higher the government spending
multiplier dydg " . On the other hand, if mark-up is procyclical (! < 0),
a more pronounced cycle elasticity (! ")lowers the expansionary e¤ects of
government purchases on output dydg # :
What does economic literature have to say about the direction of mark-up cyclicality?
Theoretical literature has mainly focused on countercyclicality,1 by taking
two alternative roads that we could label "the macroeconomic view" and "the industrial organization view".
As to the former, the traditional explanation has centered on nominal rigidities: if prices are sticky, an increase in aggregate demand - assuming
1Lindbeck and Snower (1987) and Bils (1987) achieve mark-up countercyclicality by
es-tablishing a positive relationship between aggregate demand and elasticity of demand. Ed-mond and Veldkamp (2009) assign the central role to income distribution: during booms, income shifts towards the lower tail of the distribution, featured by higher-elasticity
con-‡exibility of some elements of marginal costs - results in a mark-up reduction (Rotemberg and Woodford 1999, Woodford 2003).
The industrial organization view focuses instead on …rms’ strategic in-teraction in a non-competitive environment. Rotemberg and Saloner (1986) argue that oligopolies are likely to behave more competitively when demand rises, especially when price is the strategic variable. Under these circum-stances, in fact, the bene…t from deviation is larger, and the punishment is diminished because it will be implemented when the expansionary demand shock will have already been absorbed. As a results, price/marginal cost ratio declines as aggregate demand increases. Haltiwanger and Harrington (1991) extend the analysis to allow for time-varying …rms’expectations on future de-mand, by relaxing the assumption of i.i.d. demand shocks so to induce serial correlation in the cycle. They highlight potential asymmetries in collusive pricing behavior across di¤erent state of the business cycle, as their …ndings show that collusion is more di¢ cult during recessions than during booms. In fact, establishing a relation between future and current demand induces asymmetries between the opportunity costs of engaing in price wars accord-ing to the direction of demand. Under fallaccord-ing demand, the forgone collusive pro…ts are on a intertemporally decreasing path, so the incentive to collude is lower and likely to remain so. On the other hand, in a period of increasing demand, the traditional Rotemberg and Saloner result is mitigated by the fact that joint-maximizing pro…ts are going to be higher in the future. Along this path, Bagwell and Staiger (1997) develop a theory of collusive pricing in a framework where aggregate demand alternates stochastically between slow and fast growth states, and where the transition is governed by a Markov process. They …nd that the cyclical behaviour of collusive prices depends cru-cially on correlation of demand growth rates through time and the expected duration of boom and recessions. Particularly, collusive prices are procyclical in presence of positive demand correlation through time, and countercyclical otherwise. Furthermore, the amplitude of the collusive pricing is larger when the recession has a longer expected duration or - conversely- when the boom has a lower lenght. Those two contributions stress that the qualitative and quantitative dimensions of collusive pricing can di¤er according to the state of the business cycle. Such an asymmetry in the intensity of the collusion is directly related to the mark-up cyclical behaviour and thus - as we will argue - to the size of government spending multipliers. Therefore, those results might provide an explanation for multipliers’asymmetries over the business
cycle (Canzoneri et al., 2011)2.
More recently, a new strand of literature combines traditional general equilibrium macro models with an industrial organization approach, empha-sizing the procyclicality of entry in determining mark-up countercyclicality, through the competition e¤ect (Ghironi and Melitz 2005, Jaimovich and Floetotto 2008, Etro and Colciago 2010)
However, how empirically robust is the evidence about mark-up counter-cyclicality?
Although a considerable number of contributions points towards
coun-tercyclicality,3 empirical literature on mark-up cyclical behaviour is not
un-ambiguous. Donowitz et al. (1986,1988) …nd evidence on procyclicality in the US; Chirinko and Fazzari (1994) use a dynamic factor model to esti-mate markups, …nding that they are procyclical in nine of the eleven 4-digit industries they analyze. Updating Bils (1987) analysis - in favor of counter-cyclicality - with more recent and richer data, Nekarda and Ramey (2010) …nd that all measures of markups are either procylical or acyclical.
Hall (2009) provides a simple …rst-cut test for cyclicality by noting that the mark-up can be expressed as the ratio between the elasticity of output
with respect to labor input @Y@LLY and the share of labor compensation over
nominal income W LP Y = s . In fact, since by the envelope theorem property, a
cost-minimizing …rm equalizes the marginal cost of increasing output across all possible margins for varying production, we can express marginal cost as:
M C = W@Y
@L
(1) As gross mark-up (= ) is de…ned as the ratio between price index and
marginal costs,and multiplying and dividing by Y
L; then: = @Y @L L Y s (2)
2Rotemberg and Woodford (1992) develop the idea of mark-up countercyclicality in
a dynamic general equilibrium setting, …nding that the model’s empirical performances are closer to actual postwar US data than the corresponding predictions of the perfectly competitive model.
If the production process is approximated by a Cobb-Douglas Y = L K1 ;
then the numerator of (2) is and can be considered relatively stable over
time. Thus, the countercyclicality of the mark-up requires the
procyclical-ity of labor share s:
Figure 1 shows the dynamics of labor share and output in …ve major OECD economies from 1990 to 2009.
We can notice that labour share is far from showing an unambigous pro-cyclical behaviour ; table 1 reports a simple correlation analysis showing that - with the exception of US - there seems to be a negative rather than positive correlation between labor share and the business cycle.
Table 1: Correlation between detrendend GDP and labour share
COUNTRY CORRELATION US 0.61 UK 0 FRANCE -0.12 GERMANY -0.52 ITALY -0.49
In this paper, we present a theoretical framework based on strategic in-teraction able to rationalize the existence of pro-cyclical pricing. Our bench-mark model is the one put forward by Rotemberg and Saloner (1986). We indeed set out by o¤ering a brief summary of their analysis, reconstructing the countercyclical behavour of prices in a simple repeated duopoly game with homogeneous goods, in which demand is subject to random shocks a¤ecting the reservation price. In addition to their result, we show that increasing the probability of a positive shock brings about an increase in cartel stability, which could be large enough to more than counterbalance the e¤ect identi-…ed by Rotemberg and Saloner. Then, we review the established IO debate on the behaviour of …rms involved in an implicitly collusive price supergame under product di¤erentiation and perfect certainty, producing well de…ned procyclical conclusions. Our e¤ective contribution consists in bridging the two approaches, pursuing two distinct but related goals: (i) to characterise the maximum degree of collusion (i.e., the highest collusive price) that can be sustained in a stochastic environment, given time preferences and product di¤erentiation; and (ii) to check whether the counterciclity emerging under stochastic demand and perfect product substitutability can indeed be com-patible with the seemingly opposite result produced by the traditional cartel theory belonging to IO, in which much emphasis is posed on product di¤er-entiation but this is accompanied by perfect certainty. Our analysis indeed shows that, provided unilateral deviations grant monopoly power, Rotemberg
and Saloner’s countercyclical pricing can be seen as the limit of a general model in which, for su¢ ciently high (low) degrees of product di¤erentiation, procyclical (countercyclical) pricing obtains.
The rest of paper proceeds as follows. Section 2 recalls the Rotember and Saloner’s framework, generalized in stochastic demand framework. Sec-tion 3 inserts product di¤erentiaSec-tion and proves that this feature reverts the traditional countercyclical pricing of the benchmank model. In section 4 we provide a framework able to bridge the two positions. Finally, section 5 concludes.
2
Preliminaries: the status-quo.
We set out by brie‡y summarising …rst Rotemberg and Saloner’s (1986) model of countercyclical pricing, and then the features of implicitly collusive pricing in supergame-theoretic models deeply investigated in the industrial organization literature.
2.1
Countercyclical pricing
The following is a simpli…ed version of their setup, in which we will focus on the behaviour of a cartel formed by two (instead of n) …rms, without further
loss of generality.4 As in their paper, we consider a market for a homogenous
good, over an in…nite horizon. Time t is discrete, with t = 0; 1; 2; :::1; and
the demand function at any time t is pt = t q1t q2t;with qit 0being …rm
i’s output. Firms have identical technologies represented by the cost function
Ci = cqi; with t> c 0: To ease the exposition throughout the paper, and
again w.l.o.g., we set c = 0. The pro…t function of the individual …rm thus
coincides with revenues, i = pqi. The reservation price t is stochastic, and
in each period can take one of two values, a > b; with probabilitiesp (a) = m
and p (b) = 1 m; respectively, with m 2 [0; 1] :
The supergame unravels following the rules of Friedman’s (1971) per-fect folk theorem, whereby any unilateral deviation from the collusive path
4This exposition relies on (and slightly generalises) the simpli…ed version of the
is punished by a permanent reversal to the Nash equilibrium of the con-stituent stage game forever (the so-called grim trigger strategy). In the present setting, product homogeneity entails that the per-period pro…ts at the Bertrand-Nash equilibrium are nil. As in Rotemberg and Saloner (1986), suppose …rms (i) set prices after having observed the state of demand (either
a or b), and (ii) collude at the monopoly price in each period. Later, we will
come to the case of partial collusion.
At any t; monopoly price pMt = t=2 delivers the individual expected
cartel pro…t: E C = m M(a) + (1 m) M(b) 2 = ma2+ (1 m) b2 8 : (3)
If a …rm contemplates the possibility of deviation, the best option is to do so in a period of high demand, so that slightly undercutting the monopoly
price grants the cheating …rms full monopoly pro…ts in that period, D =
a2=4: As already explained above, such deviation at any t is punished by
driving pro…ts to zero from t + 1 to doomsday through the adoption of the marginal cost pricing rule at the Bertrand-Nash equilibrium. Assuming …rms share identical time preferences measured by a symmetric and time-invariant
discount factor 2 (0; 1) ; the stability of price collusion requires to meet
the following necessary and su¢ cient condition:
E C 1 X t=0 t D , a 2[1 (1 + m)] + b2(1 m) 8 (1 ) (4)
which is satis…ed by all
a2 a2(1 + m) + b2(1 m) ; (5) with @ @a = 2ab2(1 m) [a2(1 + m) + b2(1 m)]2 > 08m 2 [0; 1) : (6)
Property (6) indicates that the critical threshold of the discount factor stabil-ising full collusion increases with the good state. This is one of the elements leading to the (by now classical) interpretation of this model, according to which …rms should collude less if demand gets higher, as the size of the mar-ket ensures high pro…ts anyway, and this this suggests the idea of counter-ciclical pricing. This argument is reinforced if one examines the perspective
of activating some degree of partial collution at the highest pP
2 0; pM
sustainable for < :
All of the above is based on comparative statics taken w.r.t. a. Another possibility (not examined in Rotemberg and Saloner, 1986) is to assess the e¤ect of a change in m on cartel stability:
@
@m =
a2(a b) (a + b)
[a2(1 + m) + b2(1 m)]2 < 08m 2 [0; 1) (7)
revealing that an increase in the probability attached to the good state makes full collusion easier to sustain. This, on the contrary, has a de…nite procycli-cal ‡avour. Moreover, using (6-7), we can construct the marginal rate of substitution between a and m as follows:
da dm = @ @m @a @ = a (a b) (a + b) 2b2(1 m) > 0; (8)
illustrating the complementarity between a and m. Correspondingly,
yields the map appearing in Figure 1, with 1 > 2 > 3: The curves are
upward sloping and convex in m; for all m 2 [0; 1] and a > b: Hence, given any admissible value of m; an increase in a makes it more di¢ cult for …rms to sustain collusion along the frontier of monopoly pro…ts. Conversely, however, given any a > b, an increase in m generates the opposite e¤ect. Overall, the balance between the two may indeed leave the critical threshold unmodi…ed. This suggests that, if probabilities can be generated of at least a¤ected by a policy maker’s announcements, a governement might indeed exploit its own credibility to in‡uence …rms’pricing behaviour in the desired direction.
Figure 1: The critical discount factor for full collusion in the space (m; a). 6 -1 (b; 0) a m 2 3 1
We may summarise the above exposition into the following:
Lemma 1 Under demand uncertainty and product homogeneity, any positive
shock on demand increasing the level of the reservation price in the best state makes price collusion more di¢ cult to sustain, all else equal.
2.2
Implicit price collusion in oligopoly supergames
The standard approach in industrial organization theory rules out uncer-tainty to focus instead on the bearings of product di¤erentiation on the
in-tensity and stability of implicit collusion.5 Accordingly, we consider a market
where two single-product …rms o¤er di¤erentiated products over discrete time
5The material appearing in this section is a compact exposition of a large debate. See
Deneckere (1983), Majerus (1988), Chang (1991, 1992), Rothschild (1992), Ross (1992), Lambertini (1997) and Albæk and Lambertini (1998), inter alia.
t = 0; 1; 2; 3; :::1. At any t; the inverse demand function for variety i is (see Spence, 1976; and Singh and Vives, 1984, inter alia):
pi = a qi sqj (9)
where s 2 [0; 1] is the symmetric degree of substitutability between any pair of varieties. If s = 1; products are completely homogeneous; if instead s = 0; strategic interaction disappears and …rms are independent monopolists over
n separated markets. The direct demand function to be used under Bertrand
behaviour obtains by inverting the system (9):
qi = a 1 + s pi 1 s2 + spj 1 s2: (10)
As in the previous model, …rms share the same technology, summarised by
the cost function Ci = cqi; wit marginal cost c being normalised to zero for
the sake of simplicity. Therefore, per-period individual pro…ts are i = piqi:
Throughout the game, …rms also share the same intertemporal preferences
measured by the constant discount factor 2 (0; 1) :
The perfect folk theorem yields the following condition, that must be satis…ed in order for the collusive path to be stable:
C 1 X t=0 t D + N 1 X t=1 t ; (11)
where, unlike the previous model, N 0 is the pro…t generated by the
Bertrand-Nash equilibrium of the constituent game; it is positive whenever some degree of product di¤erentiation exists. Likewise, product
di¤erentia-tion entails that D
2 C; 2 C ; i.e., the cheating …rm may not necessarily
stand alone in the market in the deviation period. Four relevant cases are to be examined:
…rms’ time preferences allow for full collusion at the pure monopoly price, and product di¤erentiation is high enough to allow the cheated …rm to operate on the market with positive market share and pro…ts in the deviation period;
…rms’ time preferences allow for full collusion at the pure monopoly price, but product di¤erentiation is low enough to cause the cheated
…rms’time preferences only allow for some degree of partial collusion, and product di¤erentiation is high enough to allow the cheated …rm to operate on the market with positive market share and pro…ts in the deviation period;
…rms’time preferences only allow for some degree of partial collusion, but product di¤erentiation is low enough to cause the cheated …rm’s market share and pro…ts to fall to zero in the deviation period.
What is common to all of the above four cases is the per-period
Bertrand-Nash pro…t N appearing on the r.h.s. of (11), that can be quickly worked
out once and for all.
The …rst order condition for the maximization of i w.r.t. pi is
@ i
@pi
= a (1 s) 2pi+ spj
1 s2 = 0 (12)
whereby the symmetric Bertrand-Nash equilibrium price is
pN = a (1 s)
2 s (13)
with pro…ts
N = a2(1 s)
(1 + s) (2 s)2; (14)
both falling to zero in the homogeneous good case.
2.2.1 Full collusion
Under full collusion, both …rms set the monopoly price
pM = a
2 (15)
which yields the symmetric share of monopoly pro…ts
C = a 2 4 (1 + s) = M 2 (16) to each …rm.
Now look at the deviation period, and suppose the cheated …rms remains on the market selling a positive output. The optimal deviation is the best
reply to pM; i.e.:
pD pM = a (2 s)
4 (17)
which is viable as long as the resulting sales volume of the …rm remaining loyal to the cartel is positive, which requires
s2h0;p3 1 : (18)
The deviation pro…ts are
D = a2(2 s)
2
16 (1 s2): (19)
If instead s 2 p3 1; 1 ; then the cheated …rm is out of business and
the deviator stands alone on the market place. In this range, the deviation price solves qch = a 1 + s pM 1 s2 + spD 1 s2 = 0; (20) yielding pD0 pM = a (2s 1) 2s (21)
delivering deviation pro…ts equal to
D0
pM = a
2(2s 1)
4s2 : (22)
Then one can easily verify that
pD pM = pD0 pM and D pM = D0 pM (23)
in correspondence of s =p3 1:
We can now turn to the derivation of the critical thresholds of the discount
factor , above which full collusion is sustainable in the two cases. For
all s 2 0;p3 1 , (19) is the relevant deviation pro…t, and the critical
threshold of the discount factor above which full collusion in prices is stable is = D pM M = (2 s) 2 (24)
while for all s 2 p3 1; 1 ; i.e., in the region where (22) applies, it is 0 B = D0 pM M D0(pM) BN = (2 s)2[s (1 + s) 1] (2 s)2[s (1 + s) 1] + s4 (25) with B0 = 1=2 if s = 1: 2.2.2 Partial collusion
Here we deal with the performance of a cartel whose members’time prefer-ences are below the thresholds identi…ed above. The issue, in such a case, is
to …nd the highest collusive price p 2 pN; pM sustainable over time, given
. If all …rms set the collusive output p ; the per-period pro…ts of each cartel member are
C = (a p ) p
1 + s : (26)
Again, the optimal deviation against the cartel price will take two dif-ferent forms, depending on the degree of substitutability s. If the latter is su¢ ciently low (i.e., product di¤erentiation between the two varieties is high enough), the deviator will adopt its best reply to cartel pricing solving (12)
in which one has to plug pj = p ; to get the optimal deviation price
pD(p ) = a (1 s) + sp
2 : (27)
This generates the following one-o¤ deviation pro…ts
D(p ) = [a (1 s) + sp ]
2
4 (1 s2) ; (28)
provided that the cheated …rm is still active, i.e., it must be selling a positive
quantity qch. This happens if
qch= 1 2 a p p 2 (1 s) + 2 (a p ) + p 2 (1 + s) > 0 (29) which requires p 2 pN;a (1 s) [2 + s (6 s)] 2 s2 : (30)
In this price range, condition (11) yields the following highest collusive price
p = a (1 s) [4 s (1 ) (1 s)]
(2 s) [4 (1 s) + s2(1 )] (31)
which is monotonically increasing in a and lower than pM for all admissible
values of s and .
Now we turn our attention to the collusive price range wherein any unilat-eral deviation makes the cheating …rm a monopolist, driving all loyal cartel members out of business. This happens if expression (29) is non positive, i.e., for all
p 2 a (1 s) [2 + s (6 s)] 2 s2 ; a + c 2 (32) with a + c 2 a (1 s) [2 + s (6 s)] 2 s2 for all s 2hp3 1; 1i: (33)
Imposing qCh = 0 yields the deviation price
pD0(p ) = p a (1 s)
s (34)
which in turn delivers the deviation pro…ts
D0 pM = (a p ) [p a (1 s)]
4s2 : (35)
It can be easily checked that pD0
(p ) = pD(p )in correspondence of
p = a (1 s) [2 + s (6 s)]
2 s2 : (36)
Plugging the above expression in the stability condition (11) and solving w.r.t. p ; one obtains p 0 = a h 4 (5 2s) s2 (2 s)2(1 + s) sp i 2 (2 s) [(1 ) (1 + s) s2] (37)
in which
= 2 (4 + s) s (2 s) s3 + (2s 1) s2 4 (1 s) (4 + s) s +(2 s)2:
(38)
One can verify that > 0; by solving = 0 w.r.t. and checking that
the resulting solutions 6= R for all s 2 p3 1; 1 . Therefore, has the
same sign as the coe¢ cient of 2 in (38), which is positive, as can be easily
ascertained. Moreover, the denominator of p 0 is positive for all s and in
the unit interval.
The next step consists in observing that
a (1 s) [2 + s (6 s)] 2 s2 < p 0 < a 2 < p 0 + (39)
for all < B0: Accordingly, we select p 0 as the optimal collusive price in the
admissible parameter range identi…ed by
2 ; and s 2hp3 1; 1i: (40)
The partial derivative of p+0 w.r.t. a is:
@p 0
@a =
(2 s) [(1 ) (2 + s) s2(2 )] sp
2 (2 s) [(1 ) (1 + s) s2] (41)
Therefore, relying on the fact that > 0, we may evaluate the sign of the
numerator of @p 0=@a by evaluating the sign of
(2 s)2 (1 ) (2 + s) s2(2 ) 2 s4 (42)
which turns out to be positive for all 2 h0; e : The latter result, together
with Lemma 3, implies that @p 0=@a > 0in the entire admissible range (40).
Hence, the foregoing discussion allows us to formulate:
Lemma 2 In the Bertrand supergame under perfect certainty, any increase
(resp., decreases) in consumers’reservation price increases (resp., decreases) the intensity of partial collusion, for any degree of product di¤erentiation and irrespective of whether unilaterial deviation from the collusive path grants the cheating …rm monopoly power or not.
3
Bridging two visions
What we have reviewed so far boils down to the following two synthetic and seemingly antithetic messages:
i] if goods are undi¤erentiated and the market is subject to stochastic shoks
a¤ecting consumers’ reservation price, then …rms’ pricing behaviour exhibits a de…nite countercyclical pattern;
ii] if goods are di¤erentiated and the demand level is deterministic, then
optimal cartel prices are always monotonically related to the reservation price, heedless of …rms’time preferences.
Our aim in this section is to develop anew a model in which product di¤erentiation and uncertainty operate together, so as to see whether the above conclusions may indeed be compatible with each other. As a …rst step, we will examine the case in which duopolistic competition survives unilateral deviations from cartel pricing. The second step will be to look at the opposite case where the defecting …rm attains a monopolistic position.
On the supply side, the setup is the same as above. On the demand side, the demand function for …rm i will be:
qit = t 1 + s pit 1 s2 + spjt 1 s2; (43)
with t 2 fa; bg ; a > b > 0 and probabilities p (a) = m and p (b) = 1 m;
respectively, with m 2 [0; 1] :
3.1
Best reply deviation and the persistence of duopoly
Here, the deviation price maximises D and qch> 0: We set out by taking a
quick look at the stability condition for full collusion.
Monopoly price in state t is pMt = t=2; delivering expected per-…rm
cartel pro…ts
E C = ma
2+ (1 m) b2
4 (1 + s) : (44)
The deviation price and pro…ts in correspondence of the best demand state
The individual expected Bertrand-Nash pro…ts in each period of the punish-ment phase are given by
E N = [ma
2+ (1 m) b2] (1 s)
(1 + s) (2 s)2 : (45)
As a result, collusive stability now requires
a2(2 s)2 a2 4m (1 s) (2 s)2 + b2(1 m)2(1 s)2 ; (46) with @ @a / (1 m) (1 s) > 08m 2 [0; 1) ; s 2 h 0;p3 1 (47) and @ @m / (1 s) < 08s 2 h 0;p3 1 : (48)
As for partial collusion, de…ne
E C = m (a p (a)) p (a) + (1 m) (b p (b)) p (b)
1 + s (49)
with p (b) = b=2, in such a way that the only unknown is the partially collusive price in the best state, p (a). That is, we assume …rms will charge the best collusive price in the worst state, and appropriately tune p (a) so as to satisfy the stability condition (11), which we are about to construct step by step.
Accordingly, the best deviation against p (a) along the reaction function is
pD(p (a)) = a (1 s) + sp (a)
2 : (50)
Then the expected payo¤ in each period of the punishment phase is (45). From the usual stability condition, we get the pair of solutions:
p (a) = a (2 s) (1 s) 2 (1 s) s p (2 s) [s2(1 ) + 4 (1 s)] (2 s)2 4m (1 s) (2 s)2 (51) where = [2 m + (2 s) (1 )] s2(1 ) + 4 (1 s) > 0 (52)
and
= a2m2 s2(1 ) + 4 (1 s) 2+ (53)
4b2(2 s)2(1 ) (1 m) (2 s)2 4m (1 s) (2 s)2 > 0:
It is then easily checked that lim
m!1p (a) =
a (1 s) [4 s (1 ) (1 s)]
(2 s) [4 (1 s) + s2(1 )] = p (54)
i.e., the same price as in (31). Finally, taking the partial derivative of p (a) w.r.t. a; one can verify that
@p @a / a 2 m2 s2(1 ) + 4 (1 s) (2 s)2(1 ) + 4m + 4b2(2 s)4(1 ) (1 m) [2 m + (2 s) (1 )]2 > 0 (55) everywhere.
Therefore, we can state
Proposition 3 If deviation from the collusive path does not grant monopoly
power, then the maximum collusive price sustainable under stochastic demand conditions is monotonically increasing in the level of the best demand state.
3.2
Defecting to monopoly
The last step consists in investigating the case in which a unilateral devia-tion from the cartel price turns the deviator into a monopolist. Under full collusion, the only detail that has to be modi…ed is the deviation price in correspondence of the best state, which causes the cheated …rm’s output to drop to zero. This is (21), ensuring the deviation pro…ts (22) for all
s 2 p3 1; 1 . The resulting stability condition is
a2(2 s)2[(1 + s) s 1]
a2[s4(m + 1) + 4 (4s 1) s2(1 + 3s)] + b2(1 m) s4
0
; (56)
with @ 0=@a > 0and @ 0=@m < 0for all s 2 p3 1; 1 , so that the picture
remains much the same as we already know it, along the frontier of industry pro…ts.
Now suppose …rms’time preferences fall short of (56). If so, they may ac-tivate the highest sustainable degree of partial collusion. In such a case, they set p (b) = b=2 whenever demand is low, and solve the stability condition w.r.t. p (a) ; obtaining: p 0(a) = a (2 s) [(1 ) (2 s (2s 1)) ms 2] sp 2 (2 s) [(1 ) (1 + s (1 s)) ms2] (57) in which = a2 s2(1 (1 ms))2+ (58) 4 (1 ) ((1 ) (1 s) m (2s 1))] b2 s2(1 m) (1 ) (1 + s (1 s)) ms2 with (1 ) (2 s (2s 1)) ms2 > 0; (59) s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s) m (2s 1)) > 0 (60) and s2(1 m) (1 ) (1 + s (1 s)) ms2 < 0 (61)
for all and m in the unit interval and all s 2 p3 1; 1 : In the same
parameter region, one also has that (i) >for all
a2
b2 >
s2(1 m) [(1 ) (1 + s (1 s)) ms2]
s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s) m (2s 1)) (62)
with the threshold on the r.h.s. of (62) being always lower than one (hence,
both p 0(a)and p+0(a) are real), and (ii) p
0
(a) :
To identify the correct solution, it su¢ ces to verify that
lim m!1p 0 (a) = ah4 (5 2s) s2 (2 s)2 (1 + s) sp i 2 (2 s) [(1 ) (1 + s) s2] = p 0 : (63)
The last step consist in di¤erentiating p 0(a) w.r.t. a; de…ning a = rb
with r > 1 and then solving @p 0(a) =@a = 0 w.r.t. r; getting6
r = s (2 s) [(1 ) (2 s (2s 1)) ms
2]p (1 m) [(1 ) (1 + s (1 s)) ms2]
2p(1 s) = r
(64)
with
= (1 ) (1 + s (1 s)) ms2 ms2 (1 ) (2 s)2(1 + s)
s2(1 (1 ms))2+ 4 (1 ) ((1 ) (1 s) + m (2s 1)) : (65)
Now, it can be shown analytically that r 2 R+; moreover,
lim m!1r = 0 and (66) lim m!0r = s [2 (1 s2) + s]p (2 s)p(1 ) (1 s2) > 1 (67) for all > (2 s) 2 (1 s2) s2+ 4 [1 s (1 + s (1 s)) (1 s) (1 s2)] = (68)
which is decreasing and concave in s, with = 3 39 + 38p3 =937 ' 0:336
in s = p3 1 and = 0 in s = 1: Consequently, we have that
@p 0(a)
@a > 0 for all r > max f1; rg (69)
and conversely between one and r, if indeed r > 1: In general, r is decreasing and concave in m (this can ve ascertained numerically), and will give rise to a picture like the one reported in Figure 2, in which "+" and "-" signs
Figure 2: Collusive pricing in the space (m; r). 6 -r m 1 + (0; 0) 1
To sum up, above the upper envelope @p 0(a) =@a > 0;if r > 1 for at least
some acceptable parameter values, then in such a region @p 0(a) =@a < 0:7
An analogous exercise can be carried out in the space (s; a) as well as in
( ; a) : This is done in Figures 3 and 4, respectively. In particular, Figure 3
shows that countercyclical pricing emerges in the region in which r 2 (1; r) and the two product varieties are su¢ ciently close substitutes. It is also worth observing that r asymptotically increases to in…nity as s approaches 1 in the limit, in which case pricing is countercyclical irrespective of r; as in Rotemberg and Saloner’s (1986) original formulation.
Figure 3: Collusive pricing in the space (s; r). 6 -r s 1 + (0; 0) 1
Figure 4: Collusive pricing in the space ( ; r). 6 -r 1 + (0; 0) 1=2
Our …nal result can be therefore stated as follows:
Proposition 4 If deviation from the collusive path grants monopoly power,
then the maximum collusive price sustainable under stochastic demand condi-tions is monotonically decreasing (increasing) in the level of the best demand state if product di¤erentiation is low (high) enough.
4
Concluding remarks
The e¤ective size of government spending multipliers is a crucial macroeco-nomic issue, especially after the massive …scal stimuli of 2009-2010 and the need for spending cuts which is becoming more and more pressing. In im-perfectly competitive frameworks, multipliers’size is increasing function of mark-up countercyclicality but decreasing function of its procyclicality. As a
consequence, determining the e¤ective direction of pricing cyclicality becomes crucial for the analysis of a government spending stimulus on agggregate ac-tivity.
We have used a repeated duopoly game to revisit the issue of cyclical pricing so as to reconcile Rotemberg and Saloner’s (1986) results about the emergence of countercyclical pricing under uncertainty with the procyclical ‡avour traditionally associated with implicit collusion in the perfect certainty approach typical of the large debate on this topic in the theory of industrial organization.
The bottom line of our analysis is that the cyclical properties of …rms’ pricing behaviour are sensitive to the degree of product di¤erentiation across product varieties, in such a way that pricing is procyclical whenever the cheated …rm retains a positive market share during deviations (because prod-ucts are weak substitutes), while instead counterciclicity indeed obtains pro-vided that (i) the deviator becomes a monopolist and (ii) product di¤erenti-ation is su¢ ciently low.
Mark-up cyclical behaviour might therefore be a far more complicated issue than previously thought, with relevent consequences on the actual ef-fects of expansionary expenditure-based …scal policies. Future research will be concerned with an empirical analysis attempting to link di¤erent sectors (featured by di¤erent degrees of product di¤erentiation) with di¤erent cycli-cal properties of their average mark-ups.
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